AMO 2017 – Secondary 1 (Grade 7)

An integer must be subtracted from $-74$ to give the result $-59$. Find the square of that integer.

Write the expression $\sqrt{2^6}-\sqrt[3]{2^6}$ as an integer.

The first $5$ instances of a pattern involving the sum of three squares are shown. Find the sum $A+B+C+D$.

What is the sum of the digits in the product $2017\times9999$?

Find the value of $(1.6)^2+3.2\times0.4+0.16$.

The counting numbers are arranged five at a time, in every row, as shown. What is the least counting number in the row in which $214$ will eventually appear?

If $4096=a^b$, find the sum of all possible values of $a$ where $b$ is a whole number greater than $1$.

Clarification: use the intended domain that $a$ is a positive integer.

The spinner shown is divided into $5$ equal parts. Spin the needle once. If it lands on a perfect square number, flip a fair coin once. If it lands on a prime number, do not flip the coin. What is the probability that you flipped a coin and it landed heads up? If your answer is $p/q$ in simplest form, write $p+q$.

First, find all ordered pairs of integers $(x,y)$ satisfying $x^y=16$. Next, find the sum of all possible values of $x$ and the sum of all possible values of $y$. Finally, find the positive difference between the two sums.

Clarification: add the coordinates over all ordered pairs, counting a repeated coordinate once for each pair in which it occurs.

A survey of $40$ students at Mathlete Academy found that $27$ students play sports and $20$ students play an instrument. If $8$ students do neither, how many students play both a sport and an instrument?

$a/b$ is a fraction in simplest form, where $\sqrt{\frac1{144}-\frac1{169}}=\frac ab$. Find $a+b$.

Some of the $12$ edges of a cube are to be coloured red, so that each face has exactly $2$ red edges. What is the number of red edges?

There are exactly $n$ whole-number cubes between $8^6$ and $5^9$. Find $n$. Do not include either endpoint.

The cardboard pattern shown is composed of six numbered squares and is folded to form a cube. What is the sum of the digits in the greatest product of the numbers on three faces that meet at a common vertex?

How many $4$-digit odd integers can be formed from the digits $0,1,4,6,8$ if no digit can be repeated?

Two squares with integer side lengths overlap so that two sides of the smaller square rest along two sides of the larger square, as shown. The L-shaped shaded region has area $21$ square units. Find the sum of the two possible side lengths of the smaller square.

Each of $10$ cards displays two different symbols, one on the front and the other on the back. The symbols are clubs, diamonds, spades, hearts and a dot. The front of each card is shown. No two cards have the same pair of symbols. What is the least number of cards you can turn over to ensure that a dot will appear?

For how many whole numbers from $5001$ to $5499$ does the product of the middle two digits exceed $6$?

Jethro wishes to tile a $10$ cm by $15$ cm rectangular floor using giant J-shaped tiles. Each tile has area $6\text{ cm}^2$ and is composed of $1$ cm by $1$ cm square sections as shown. What is the smallest area, in square centimetres, of the floor that cannot be covered? Tiles cannot be cut or overlapped.

Mel sells jumbo cheesy pretzel nuggets in a $4$-nugget diet size, a $7$-nugget share size and a $17$-nugget party size. What is the greatest total number of nuggets that cannot be purchased? For example, $8$ nuggets can be purchased, but $9$ cannot.